Numerical techniques for solving the inverse retrospective problem of thermal evolution of the Earth interior

We consider an inverse (time-reverse) problem of thermal evolution of a viscous inhomogeneous incompressible heat-conducting fluid describing dynamics of the Earth’s mantle. Present observations of geophysical fields (temperature, velocity) are incorporated in a three-dimensional dynamic model to determine the initial conditions of the fields. We present and compare numerical techniques for solving the inverse problem: backward advection, variational (adjoint), and quasi-reversibility methods. The methods are applied to restore the evolution of the mantle structures such as rising plumes and descending lithospheric plates.